HIGH-ORDER ITERATIVE METHODSFOR A NONLINEAR KIRCHHOFF WAVE EQUATION

摘要

In this paper we consider the following nonlinear wave equation u_(tt) — θ/(θx)(u(x,t,||u_x(t)||~2)u_x) = f(x, t, u), 0 < x <1, 0 < t < T, {u(0,t) = u(1,t) = 0, u(x, 0) = u_0(x), u_t(x, 0) = u_1(x), where u, f,u_0,u_1 are given functions satisfying conditions specified later. In Eq. (1)_1, the nonlinear term μ(x, t, ||u_x||~2) depends on the integral ||u_x||(t)||~2 =∫_0~l|u_x(x,t)|~2 dx. In this paper we associate with equation (1)_1 a recurrent sequence {u_m} defined by (θ~2u_m)/θt~2-θ/θx(μ(x,t,||u_(mx)(t)||~2)u_(mx)=Sun from i=0 to N-1 of 1/i (θ~if)/(θu~i)(x,t,u_(m-1)(u_m-u_(m-1))~i, 0 < x < 1, 0 < t < T, with u_m satisfying (1)_(2,3). The first term u_0 is chosen as u_0≡ 0. If f ∈ C~N ([0, 1] × R_+ × R.), we prove that the sequence {u_m} converges at a rate of order N to a unique weak solution of problem (1).